Requisites

Additional Requirements

Students are not permitted to take more than one of MATH37001 or MATH47201 for credit in the same or different undergraduate year.

Students are not permitted to take MATH47201 and MATH67201 for credit in an undergraduate programme and then a postgraduate programme.

Note that MATH67201 is an example of an enhanced level 3 module as it includes all the material from MATH37001

When a student has taken level 3 modules which are enhanced to produce level 6 modules on an MSc programme taken within the School of Mathematics, then they are limited to a maximum of two such modules (with no alternative arrangements available otherwise)

Aims

To provide a firm grasp of a range of basic concepts and fundamental results in the theory of martingales and to give some simple applications in the rapid developing area of financial mathematics.

Overview

An introduction to a circle of ideas and fundamental results of the theory of martingales, which play a vital role in stochastic calculus and in the modern theory of finance.

Learning outcomes

On successful completion of this course unit students will

have a good understanding of the basic concept of integration with respect to a probability measure and the basic properties of fair games;

Syllabus

Conditional expectations. Fair games and martingales, submartingales and supermartingales. Doob decomposition theorem. Stopping times and the optional sampling theorem. The upcrossing inequality and the martingale convergence theorem. The Doob maximal inequality and the martingale modification theorem. [13]

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.